Visualising Solid Shapes – Notes for CBSE Class 7 Maths

Introduction to Visualising Solid Shapes

Visualising Solid Shapes ,In our daily lives, we encounter various objects such as books, balls, boxes, and cylinders. These objects are not flat; they have length, breadth, and height, making them three-dimensional (3D). Chapter 15 of Class 7 Maths introduces us to solid shapes, their properties, and how we can visualize and represent them. This chapter builds on the understanding of two-dimensional (2D) shapes from earlier classes and extends it to 3D objects.

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1. Understanding Dimensions

• One Dimension (1D): A line has only length. It has no width or height. Example: A straight thread.
• Two Dimensions (2D): Shapes like squares, circles, and triangles have length and breadth but no height. These are flat shapes. Example: A sheet of paper.
• Three Dimensions (3D): Solid objects like cubes, spheres, and cylinders have length, breadth, and height. These are called solid shapes. Example: A football.

Solid shapes occupy space and have volume, unlike 2D shapes, which only have area.

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2. What Are Solid Shapes?

Solid shapes are 3D objects that we can touch and feel. They have:

• Faces: Flat surfaces that form the boundary of the solid.
• Edges: Lines where two faces meet.
• Vertices: Points where edges meet (corners).

Examples of solid shapes:Visualising Solid Shapes

• Cube (e.g., a dice)
• Cuboid (e.g., a book)
• Cylinder (e.g., a water bottle)
• Cone (e.g., an ice-cream cone)
• Sphere (e.g., a ball)

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3. Types of Solid Shapes:Visualising Solid Shapes

Solid shapes are broadly classified into two categories:

1. Polyhedrons: Solids with flat faces that are polygons (e.g., cube, cuboid, prism, pyramid).
   • A cube has 6 square faces.
   • A cuboid has 6 rectangular faces.
   • A triangular prism has 5 faces (2 triangles and 3 rectangles).
   • A tetrahedron (a type of pyramid) has 4 triangular faces.
2. Non-Polyhedrons: Solids with curved surfaces (e.g., sphere, cylinder, cone).
   • A sphere has one curved surface and no flat faces.
   • A cylinder has 2 flat circular faces and 1 curved surface.
   • A cone has 1 flat circular face and 1 curved surface.

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4. Properties of Common Solid Shapes:Visualising Solid Shapes

Here’s a table summarizing the properties of some common solid shapes:

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5. Euler’s Formula

For polyhedrons, there’s a special relationship between the number of faces (F), edges (E), and vertices (V). This is called Euler’s Formula:

• F + V – E = 2

Examples:

1. Cube:
   • F = 6, V = 8, E = 12
   • 6 + 8 – 12 = 14 – 12 = 2 (True)
2. Cuboid:
   • F = 6, V = 8, E = 12
   • 6 + 8 – 12 = 2 (True)
3. Triangular Prism:
   • F = 5, V = 6, E = 9
   • 5 + 6 – 9 = 11 – 9 = 2 (True)

Euler’s Formula helps verify if a given set of F, V, and E values represents a valid polyhedron.

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6. Visualising Solid Shapes

To understand solid shapes better, we use different methods to represent them on paper (which is 2D). These methods include:

1. Oblique Sketches:
   • A simple way to draw 3D shapes without using exact measurements.
   • Lines are drawn at an angle to show depth.
   • Example: Drawing a cube by sketching squares with slanted lines connecting them.
2. Isometric Sketches:
   • A more accurate method using isometric dot paper.
   • All edges are drawn at 30° angles to the horizontal.
   • It gives a realistic 3D view while maintaining equal lengths for equal edges.
3. Nets:
   • A net is a 2D shape that can be folded to form a 3D solid.
   • Example: A cross-shaped net with 6 squares can be folded into a cube.
   • Different solids have different nets (e.g., a cuboid can have multiple net arrangements).

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7. Views of Solid Shapes

When we look at a 3D object from different angles, we see different 2D shapes. These are called views:

• Front View: What you see from the front.
• Side View: What you see from the side.
• Top View: What you see from above.

Example: A Cylinder

• Front View: Rectangle
• Side View: Rectangle
• Top View: Circle

These views help us visualize and draw solid shapes from different perspectives.

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8. Shadows of Solid Shapes

When light falls on a solid shape, it casts a shadow. The shape of the shadow depends on the object and the direction of light.

• A cube can cast a square or hexagonal shadow depending on its position.
• A cylinder can cast a rectangle or circle shadow.
• A sphere always casts a circular shadow.

This concept connects geometry to real-life observations.

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9. Drawing Solid Shapes

To draw solid shapes on paper:

1. Start with basic 2D shapes (e.g., squares for a cube).
2. Add depth by drawing additional lines (oblique or isometric).
3. Practice with nets to understand how faces connect.

Steps to Draw a Cube (Oblique Sketch):

1. Draw a square for the front face.
2. Draw another square slightly offset behind it.
3. Connect the corresponding vertices with slanted lines.

Steps to Draw a Cylinder:

1. Draw two parallel circles (top and bottom faces).
2. Connect the edges with curved lines to show the surface.

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10. Key Points to Remember

• 3D shapes have length, breadth, and height.
• Polyhedrons have flat polygonal faces; non-polyhedrons have curved surfaces.
• Use Euler’s Formula (F + V – E = 2) to check polyhedrons.
• Nets help visualize how a solid is formed from a flat shape.
• Views (front, side, top) and shadows depend on the angle of observation.

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11. Solved Examples

1. Question: Verify Euler’s Formula for a tetrahedron (4 faces, 4 vertices, 6 edges).
   • F = 4, V = 4, E = 6
   • F + V – E = 4 + 4 – 6 = 8 – 6 = 2
   • Formula holds true.
2. Question: What is the top view of a cone?
   • The top view of a cone is a circle (its base).
3. Question: How many nets can a cube have?
   • A cube can have 11 different nets (various arrangements of 6 squares).

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12. Practice Questions

1. Draw an oblique sketch of a cuboid.
2. Find F, V, and E for a square pyramid and verify Euler’s Formula.
3. What are the front, side, and top views of a cube?
4. Draw a net for a cylinder.

Chapter 15 Visualising Solid Shapes